Coprimality Density Above Dimension One

MrlyProd

First published 2026-08-23, revised 2026-09-08

Stand at the origin of a square grid and look out: you can see a lattice point exactly when its coordinates share no common factor, and $6/\pi^2$ of all lattice points are visible. Now do the same on a fractal - the Sierpinski gasket, say, built by keeping the points of $[0,2^n)^2$ whose binary digit pairs all avoid $(1,1)$ . The answer changes to $16/(3\pi^2)$ , and this paper proves why, for every fractal of this kind whose dimension exceeds one. Every prime away from the base keeps its classical factor; every prime dividing the base is deleted and replaced, collectively and exactly, by a single rational number read straight off the digit set.

The level-5 Sierpinski gasket design: 243 points, the 122 visible from the origin filled and the rest hollow.

Fix a base $q \ge 2$ , a dimension $D \ge 2$ , and a set $F \subseteq \{0,\dots,q-1\}^D$ of admissible digit vectors, $k = |F|$ . The level- $n$ set $S_n$ holds the $k^n$ points of $[0,q^n)^D$ all of whose base- $q$ digit vectors lie in $F$ , and $A(n)$ counts those with coordinate gcd $1$ . The design is spanning when the differences $F - F$ generate $\mathbb{Z}^D$ , and its attractor has dimension $\log_q k$ . The bracket is $B(F) = \sum_{e \mid \mathrm{rad}(q)} \mu(e) k_e / k$ , where $k_e$ counts the corners all of whose coordinates are divisible by $e$ .

Theorem. If the design is spanning and $k > q$ - equivalently, the dimension $\log_q k$ exceeds $1$ - then

\frac{A(n)}{k^n} \longrightarrow B(F)\prod_{p \nmid q}\left(1-p^{-D}\right) = \frac{B(F)}{\zeta(D)}\prod_{p \mid q}\left(1-p^{-D}\right)^{-1}.

The proof is an elementary sieve. A digit-box bound gives $\#\{x \ne 0 \in S_n : m \mid x_i \ \forall i\} \le (q+1)^D k^n m^{-\log_q k}$ ; summed against the von Mangoldt weight it converges precisely when $\log_q k > 1$ , which is where the hypothesis lives and the only place it is used. Spanning enters once, in a character contraction that equidistributes $S_n$ modulo every $d$ coprime to $q$ . The hypothesis $k > q$ cannot stand alone: at $q = 3$ , $F = \{0,2\}^2$ has $k = 4 > 3$ yet $A(n) = 0$ forever - though one shear by $2$ restores the constant $81/(16\pi^2)$ . For the gasket the theorem settles the convergence conjecture recorded in OEIS A396934, that $a(n)/3^n \to 16/(3\pi^2)$ . The scripts re-check the finite claims: ten designs enumerated exhaustively over 67 level rows and 1,352,989 points, the gasket to $A(11) = 95260$ ; the base-6 identity that $8^n/2$ points have gcd coprime to $6$ while the naive product of per-prime marginals predicts $0.46875 \cdot 8^n$ ; the box bound in 1062 exact cases; and the exhaustive base-2, $D = 4$ census - all 65536 designs closing into 402 orbits, 336 of them inside the theorem.

Grew from the coprime page of the MrlyMath tree.
paper.pdf - the paper.
tectonic paper.tex rebuilds it; python3 scripts/verify.py re-checks every number in about three seconds; python3 scripts/figure.py redraws the gasket.